Abstract
We give a direct proof of sandwich-type theorems for linear invariant partially ordered vector space operators in the setting of convexity. As consequences, we deduce equivalence results between sandwich, Hahn-Banach, separation and Krein-type extension theorems, Fenchel duality, Farkas and Kuhn-Tucker-type minimization results and subdifferential formulas in the context of invariance. As applications, we give Tarski-type extension theorems and related examples for vector lattice-valued invariant probabilities, defined on suitable kinds of events.
Highlights
Sandwich and Hahn-Banach-type theorems are deeply studied, have several applications in different areas, and are related with several topics, among which, for example, minimization of functionals and operators
We proved some versions of Hahn-Banach and sandwich-type theorem related to convex subinvariant operators, taking values in a partially ordered vector space R, in the setting of convexity, using a technique of construction of invariant finitely additive R-valued means, which allows to find invariant linear R-valued operators from linear operators which are not necessarily invariant, whose existence is guaranteed by the classical results
We deduced some Fenchel duality-type theorem, subdifferential formulas, Krein-type extension theorems, and showed that all these results are equivalent to separation, Farkas and Kuhn-Tucker-type minimization theorems
Summary
Sandwich and Hahn-Banach-type theorems are deeply studied, have several applications in different areas, and are related with several topics, among which, for example, minimization of functionals and operators In these cases, often it is advisable to associate to a “primal” problem a “dual” problem, which is in general easier to handle, and to investigate the relations existing between them. Many studies about these topics have been extended to the context of partially ordered space-valued operators and measures (see [8,10,14,24,32] and their bibliographies) Another field of research related with sandwich-type theorems are extension theorems for finitely additive (and invariant) measures and probabilities, for example in exchangeable processes (see [4,28,39]). The invariance allows to “replace” the study of the involved events related to some periods of time about which one has not enough knowledge with the corresponding events associated with other periods of time on which one has more informations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
